my solution:lfdahl said:Given a circle (radius $R$) with an inscribed square. Now inscribe a new circle in the square and then again a new square in the new circle etc. What is the total area of the infinite number of inscribed squares?
I like Serena said:Just for illustration purposes. (Smile)
\begin{tikzpicture}[very thick]
\newcommand\Square[1]{+(-#1,-#1) rectangle +(#1,#1)}
\draw[green] foreach \r in {0,...,16} { circle ({5*2^(-\r/2)}) };
\draw[blue!50] foreach \r in {1,...,16} { \Square{{5*2^(-\r/2)}} };
\fill circle (0.08);
\end{tikzpicture}
Albert said:my solution:
the area of the 1st square=$2R^2$
the area of the 2nd square=$R^2$
the area of the 3rd square=$\dfrac {R^2}{2}$
the area of the 4th square=$\dfrac {R^2}{4}$
so the total area =$2R^2+R^2+\dfrac {R^2}{2}+\dfrac {R^2}{4}+------=4R^2$
Inscribed squares refer to a series of squares that are nested within one another, with each square sharing a side with the previous square and having its corners touching the sides of the larger square.
No, it is not possible to calculate the exact total area of an infinite number of inscribed squares as the area would approach infinity and cannot be expressed as a finite number.
Yes, there is a formula to calculate the total area of a given number of inscribed squares. It is A = s2 + (s2/2) + (s2/4) + ... , where s is the side length of the largest square.
The total area of the inscribed squares approaches infinity as the number of squares increases, but the rate of increase becomes smaller as the squares get smaller in size.
Yes, the concept of inscribed squares can be applied to other shapes such as circles, triangles, and regular polygons. The total area can also be calculated using different equations depending on the shape of the inscribed figure.